[1]} point out that, to satisfy detailed balance, one must verify that the same proposal scheme is able to return from the proposed value to the initial point. It is here that we found an error in Zappa's algorithm. [1]} use a vector space argument to find the point \(x^{\prime }\)  on \(\mathcal {T}_y\)  directly “above" the point \(x\) , in the direction of the normal vectors at \(y\) . However, they do not consider the possibility that there is another point on the manifold between \(x^{\prime }\)  and \(x\) . They propose a projection scheme that only checks whether the reverse projection succeeds in returning to the manifold, not whether it succeeds in returning to the initial point. It should be pointed out that this is a computational error, not an error in their mathematics.
